Functions, inverse functions and composite functions

Functions, inverse functions and composite functions

By Aqil Mohammed

Learning Objectives:

•To be able to find the value of a function, 𝑓 for a given 𝑥

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•To be able to find the inverse of an invertible function, 𝑓 ∶ 𝑥 → 𝑦, by changing the subject from 𝑦 to 𝑥.

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•To be able to form a composite function 𝑓(𝑔(𝑥)) using two given functions 𝑓 and 𝑔.

Starter

Multiple Choice

Select the missing number

• 6

• 7

• 8

• 9

Multiple Select

Select all possible functions for this function machine

• $$+2x$$

• $$\times3$$

• $$\times3x$$

• $$2$$

Multiple Choice

What does the function machine do?

• $$\times y$$

• $$-y$$

• $$+y$$

• $$\div y$$

Multiple Choice

Select the correct function

• -3

• +3

• -2

• +2

Multiple Choice

Select the correct function

• $$\times2$$  

• $$+a$$  

• $$2$$  

• $$+b$$  

Multiple Choice

Select the correct function

• $$\div2$$

• $$-x$$

• $$-2$$

• $$-5$$

Fill in the Blanks

Q1) g(x) = 4x + 7

a) Find the value of g(4)

Fill in the Blanks

Q1) g(x) = 4x + 7

a) Find the value of g(-1)

Fill in the Blanks

Q2) h(x) = 6 - 3x

a) Find the value of h(2)

Fill in the Blanks

Q2) h(x) = 6 - 3x

b) Find the value of h(-3)

Fill in the Blanks

Q3) $f\left(x\right)=2x^2+4$  

a) Find the value of f(3)

Fill in the Blanks

Q3) $f\left(x\right)=2x^2+4$  

b) Find the value of f(-5)

Multiple Choice

Q1) g(x) = 4x + 7

a) Find the value of $$g^{-1}\left(x\right)$$  

• $$g^{-1}\left(x\right)=\frac{x+7}{4}$$  

• $$g^{-1}\left(x\right)=\frac{x-7}{4}$$  

• $$g^{-1}\left(x\right)=7x\ -\ 4$$  

• $$g^{-1}\left(x\right)=4\ -\ 7x$$  

Fill in the Blanks

Q1) g(x) = 4x + 7

b) Find the value of $g^{-1}\left(3\right)$  

Multiple Choice

Q2) h(x) = 6 - 3x

a) Find the value of $$h^{-1}\left(x\right)$$  

•   $$h^{-1}\left(x\right)=6\ -\ \frac{x}{3}$$  

•   $$h^{-1}\left(x\right)=2\ +\frac{x}{3}$$  

• $$h^{-1}\left(x\right)=2\ -\ \frac{x}{3}$$  

•   $$h^{-1}\left(x\right)=3x-6$$  

Fill in the Blanks

Q2) h(x) = 6 - 3x

b) Find the value of $h^{-1}\left(-3\right)$  

Multiple Choice

Q3) $$f\left(x\right)=2x^2\ +4$$  

a) Find the value of $$f^{-1}\left(x\right)$$  

• $$f^{-1}\left(x\right)=\sqrt[]{\frac{x-4}{2}}$$    

•    $$f^{-1}\left(x\right)=\sqrt[]{\frac{x+2}{4}}$$  

• $$f^{-1}\left(x\right)=4-2x^2$$   

Multiple Choice

Q3) $$f\left(x\right)=2x^2\ +4$$  

b) Find the value of $$f^{-1}\left(36\right)$$  

• 4

• -4

• 4 or -4

Composite Functions

Multiple Choice

Q1) f(x) = 4 - 2x and g(x) = x + 5

a) Find the composite function f(g(x)).

• -2x - 6

• -2x + 14

• -2x + 9

• 2x - 9

Fill in the Blanks

Q1) f(x) = 4 - 2x and g(x) = x + 5

b) Find the composite function f(g(2)).

Multiple Choice

Q1) f(x) = 4 - 2x and g(x) = x + 5

c) Find the composite function g(f(x)).

• -2x - 6

• -2x + 14

• -2x + 9

• 2x - 9

Fill in the Blanks

Q1) f(x) = 4 - 2x and g(x) = x + 5

d) Find the composite function g(f(-1)).

Multiple Choice

Q2) f(x) = 2x + 1 and g(x) = $$x^2+2$$  

a) Find the composite function f(g(x)). 

• $$f\left(g\left(x\right)\right)=4x^2+2x\ +\ 4$$  

• $$f\left(g\left(x\right)\right)=4x^2+2x\ +3$$  

• $$f\left(g\left(x\right)\right)=2x^2+3$$  

• $$f\left(g\left(x\right)\right)=2x^2+5$$  

Fill in the Blanks

Q2) f(x) = 2x + 1 and g(x) = $x^2+2$  

b) Find the composite function f(g(2)). 

Multiple Choice

Q2) f(x) = 2x + 1 and g(x) = $$x^2+2$$  

c) Find the composite function g(f(x)). 

• $$g\left(f\left(x\right)\right)=4x^2+2x\ +\ 4$$  

• $$g\left(f\left(x\right)\right)=4x^2+2x\ +3$$  

• $$g\left(f\left(x\right)\right)=2x^2+3$$  

• $$g\left(f\left(x\right)\right)=2x^2+5$$  

Fill in the Blanks

Q2) f(x) = 2x + 1 and g(x) = $x^2+2$  

d) Find the composite function g(f(-1)). 

Multiple Select

Tick the topics you understand

• Finding the value of a function

• Finding the inverse of a function

• Finding the composite of a function

Poll

How do you feel about this lesson?